One Dimensional Motion – Velocity & Acceleration

Lecturer: Professor Stephen T. Thornton

Reading Quiz: The red and black lines represent the motion of two cars. Which of the following statements is most true?

A) Red and black car have the same acceleration.

B) Red car always has a higher speed.

C) Black car always has a higher speed.

D) Black car always has a lower acceleration.

E) Red car always has a lower acceleration.

Answer: E

The red car has a constant velocity and zero acceleration. Only the black car has acceleration, and it is positive.

Last time

Went through class procedures, WebAssign, etc.

Discussed length, mass, and time.

Significant figures, scientific notation.

Problem solving.

Today

Distance, displacement.

Average speed and average velocity.

Instantaneous velocity

Acceleration: average, instantaneous

Motion with constant acceleration

Copyright © 2009 Pearson Education, Inc.

Reference Frames and DisplacementAny measurement of position, distance, or speed must be made with respect to a reference frame.

For example, if you are sitting on a train and someone walks down the aisle, the person’s speed with respect to the train is a few miles per hour, at most. The person’s speed with respect to the ground is much higher.

One-Dimensional Coordinate System

DisplacementDefinition:

displacement = change in position

= final position – initial position

Δx = xf - xi

Distance = total length traveled

One-Dimensional Motion Along the x-axis

What is the total distance traveled (0-4s)?

What is the displacement? Do quiz next.

start

finish

Conceptual Quiz: What is the total distance traveled?

A) - 8 m

B) -4 m

C) -2 m

D) +4 m

E) +8 m

start

finish

Conceptual Quiz: What is the total distance traveled?

start

finish

A) - 8 m

B) -4 m

C) -2 m

D) +4 m

E) +8 m

Conceptual Quiz: What is the total displacement?

finish

start

A) - 8 m

B) -4 m

C) -2 m

D) +4 m

E) +8 m

finish

start

A) - 8 m

B) -4 m

C) -2 m Δx = xf – xi = (-1 m) – (1 m) = -2 m

D) +4 m

E) +8 m

Conceptual Quiz: What is the total displacement?

Average Speed

distanceaverage speed =

elapsed time

Note that this is always a positive number.

Average Velocity

Velocity is different than speed, because velocity is a vector.

displacementaverage velocity =

elapsed time

f iav

f i

x xxv

t t t

One-Dimensional Motion Along the x-axis

What is average velocity from 0 to 4 s?

( 1m) - (1m) 2m0.5 m/s

4s-0s 4sf i

avf i

x xxv

t t t

start

finish

direction?

Motion along the x axis represented with an x-versus-t graph (same motion as before)

Average Velocity on an x-Versus-t Graph

av 0xvt

Look at average from point A to B

Average Velocity on an x-versus-t Graph

av 0xvt

Now look from 2 s to 3 s

Instantaneous Velocity

0

dlim

dt

x xv

t t

In this way we can find the velocity at any particular instant of time.

What is the instantaneous velocity at t = 1 s?

motion

Constant Velocity Graph

Instantaneous velocity is equal to average velocity when velocity is constant.

straight line

Graphical Interpretation of Average and Instantaneous Velocity

Observable Universe. Recent findings in astrophysics suggest that the observable Universe can be modeled as a sphere of radius R = 13.7 x 109 light-years with an average mass density of about 1 x 10-26 kg/m3 where only about 4% of the Universe’s total mass is due to “ordinary” matter (such as protons, neutrons, and electrons). Use this information to estimate the total mass of ordinary matter in the observable Universe.

151 lightyear 9 46 10 m. .

Airplane Motion. An airplane travels 3100 km at a speed of 720 km/h, and then encounters a tailwind that boosts its speed to 990 km/h for the next 2800 km. What was the total time for the trip? What was the average speed of the plane for this trip?

AccelerationJust like velocity is given by the rate of change of position with respect to time, the acceleration is given by the rate of change of velocity with respect to time.

We are still dealing with one-dimensional motion, so vector direction is simple.

We use signs to denote the directions of both velocity and acceleration along a particular axis. x

When v is +, motion is to right. When v is -, motion is to left. When motion is to the right, and a is +, then object speeds up (accelerates) to the right (case II). When motion is to the left, and a is +, then object is slowing down and will eventually turn to the right. (not shown here)

Graphical Interpretation of Average and Instantaneous Acceleration

If you don’t remember about tangents, please review!

Average Acceleration

f iav

f i

v vva

t t t

We must be very careful with units. What are they?

m/s2

Run physlet

http://physics.bu.edu/~duffy/semester1/semester1.html

Constant velocity, constant acceleration

Play with these physlets at home and try the different cases. Time is too short here.

Instantaneous Acceleration

2

20

d d d dlim

d d d dt

v v x xa

t t t t t

0

dlim

dt

x xv

t t

Similarity between velocity and acceleration is clear.

Conceptual Quiz: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true?

A) At time tB, both trains have the same velocity.B) Both trains speed up all the time. C) Both trains have the same velocity at some time before tB.D) Somewhere on the graph, both trains have the same acceleration.

time

Conceptual Quiz: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true? A) At time tB, both trains have the same velocity.B) Both trains speed up all the time. C) Both trains have the same velocity at some time before tB. D) Somewhere on the graph, both trains have the same acceleration.

time

Motion with Constant Acceleration

If the acceleration is constant, then we have

where v = v0 at t = 0. This result is easy to show from our definition of a.

0v v at

The Average Velocity

constant accelerationiv

fv

Constant accelerationAnother important result:

Note that I have used vi and vf , which is more general than using v0 and v, because we may want to find the average between some initial and final position other than v0 and a general v.

Let’s determine some important equations.

av 0

1 1( )

2 2 i fv v v v v

0av

0av

av 0

0 av

0

or

solve for :

f i

f i

x x x xxv

t t t t

x xv

tv t x x

x x x v t

Insert our previous result for vav

0 0

1( )

2x x v v t

Only for constant acceleration!!!!!

0 0

1( )

2x x v v t

This is a very important equation. It relates the position x to the velocity v as a function of time t.

But we also know the relationship between velocity v and acceleration a. It was

0 constantv v a t

0 0

0 0 0 constant

20 0 constant

1( )

21

( )2

1

2

x x v v t

x x v v a t t

x x v t a t

0

0

0 0

00 0

2 22 20

0 0 0

solve for ; is constant

1We had ( )

2Substitute in for from above

1( )

2

or 22

v v at t a

v vt

a

x x v v t

t

v vx x v v

a

v vx x v v a x x

a

20 0

1

2x x v t at

av 0

1 1( )

2 2 i fv v v v v

0v v at

We now have all the equations we need to solve constant-acceleration problems.

2 20 02v v a x x

Make sure you look carefully at the examples in the text. I will not go through those.

You drive for 30 minutes at 30

mi/hr and then for another 30

minutes at 50 mi/hr. What is your

average speed for the whole trip?

A) more than 40 mi/hr

B) equal to 40 mi/hr

C) less than 40 mi/hr

Conceptual QuizConceptual QuizII

You drive for 30 minutes at 30

mi/hr and then for another 30

minutes at 50 mi/hr. What is your

average speed for the whole trip?

A) more than 40 mi/hr

B) equal to 40 mi/hr

C) less than 40 mi/hr

It is 40 mi/hr in this case. Because the average speed is distance/time

and you spend the same amount of time at each speed, then your

average speed would indeed be 40 mi/hr.

Conceptual QuizConceptual Quiz

You drive 4 miles at 30 mi/hr and

then another 4 miles at 50 mi/hr.

What is your average speed for

the whole 8-mile trip?

A) more than 40 mi/hr

B) equal to 40 mi/hr

C) less than 40 mi/hr

Conceptual QuizConceptual Quiz

You drive 4 miles at 30 mi/hr and

then another 4 miles at 50 mi/hr.

What is your average speed for

the whole 8-mile trip?

A) more than 40 mi/hr

B) equal to 40 mi/hr

C) less than 40 mi/hr

It is not 40 mi/hr! Remember that the average speed is distance/time.

Because it takes longer to cover 4 miles at the slower speed, you are

actually moving at 30 mi/hr for a longer period of time! Therefore,

your average speed is closer to 30 mi/hr than it is to 50 mi/hr.

Conceptual QuizConceptual Quiz